On the classification of group invariant solutions of the Barenblatt–Gilman model by a one-dimensional system of subalgebras

The Barenblatt–Gilman (BG) equation, which simulates nonequilibrium countercurrent capillary impregnation, is discussed in this study. By applying symmetry classification to the nonlinear function $\Phi \left(\sigma \right)$, six distinct cases emerge. In the general case, $\Phi \left(\sigma \right)$ yields a three-dimensional principal algebra. The other cases extend this Lie algebra to infinite dimensions, which are then reformulated as six-dimensional Lie algebras. For each of these six possible Lie algebras, a system of one-dimensional subalgebras is derived using P. Olver’s method. Group invariant solutions are obtained by performing symmetry reductions under the derived optimal system. The conservation laws of this model are determined using the direct (multiplier) approach. First, the multipliers based on dependent and independent variables are determined and after that, conserved vectors are constructed to correspond to these multipliers. This study presents analytical results in the form of invariant solutions, which are novel due to the nonlinearity of the function $\Phi \left(\sigma \right)$. Since very few analytical methods address such nonlinear problems, these solutions offer unique insights. Researchers focusing on numerical solutions can also utilize these results for comparative analysis.